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Null Broadening With Snapshot-Deficient Covariance Matrices in Passive Sonar H. Song, Member, IEEE, W. A. Kuperman, W. S. Hodgkiss, Member, IEEE, Peter Gerstoft, and Jea Soo Kim
Abstract—Adaptive-array beamforming achieves high resolution and sidelobe suppression by producing sharp nulls in the adaptive beampattern. Large-aperture sonar arrays with many elements have small resolution cells; interferers may move through many resolution cells in the time required for accumulating a full-rank sample covariance matrix. This leads to “snapshot-deficient” processing. In this paper, the null-broadening technique originally developed for an ideal stationary problem is extended to the snapshot-deficient problem combined with white-noise constraint (WNC) adaptive processing. Null broadening allows the strong interferers to move through resolution cells and increases the number of degrees of freedom, thereby improving the detection of weak stationary signals. Index Terms—Covariance matrix taper (CMT), null broadening, robust adaptive beamforming, snapshot-deficient processing, white-noise constraint (WNC).
I. INTRODUCTION
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ECENT trends in passive sonar systems include the use of large-aperture arrays with many elements to form narrow beams in order to detect quiet targets in a noisy background [1], [2]. This paper is concerned with the detection of weak sources in the presence of fast-moving strong interferers crossing many resolution cells in a time interval too small to build a full-rank covariance matrix. To achieve this, we combine the null-broadening approach developed for an ideal stationary problem [3]–[5] with white-noise constraint (WNC) adaptive processing [6]. At low frequencies, the background often is dominated by loud and fast surface ships that move through many narrow beams or cells in the time it takes to obtain a satisfactory sample covariance matrix. Larger arrays require longer duration snapshots due to the longer transit time of sound across the array. More snapshots are also needed due to the many elements [7]–[10]. Usually, this leads to “snapshot-deficient” processing [1]. A number of techniques have been developed to carry Manuscript received September 27, 2002; revised April 29, 2003. This work was supported by the Office of Naval Research (ONR) and Defense Advanced Research Projects Agency. This research was motivated by our participation in the ONR Ocean Acoustic Observatory Panel. H. Song, W. A. Kuperman, W. S. Hodgkiss, and P. Gerstoft are with the Marine Physical Laboratory, Scri Gerstoft are with the Marine Physical Laboratory, Scripps Institution of Oceanography, La Jolla, CA 92093-0238 USA (e-mail:
[email protected] Institution of Oceanography, La Jolla, CA 92093-0238 USA (e-mail:
[email protected];
[email protected];
[email protected];
[email protected]). J. S. Kim is with the Division of Ocean System Engineering, Korea Maritime University, Pusan 606-791, South Korea (e-mail:
[email protected]). Digital Object Identifier 10.1109/JOE.2003.814055
out adaptive processing with less-than-full-rank covariance matrices. The two most common are diagonal loading [11] and subspace methods [12], [13]. Recently, a multirate adaptive beamforming (MRABF) approach was proposed by Cox [2], which uses only a few snapshots to estimate and null the loud moving interferers, followed by more-standard adaptive procedures using many more snapshots to find weak stationary targets. Null broadening can provide a simple and robust approach to the snapshot-deficient problem arising from the motion of strong interferers when combined with robust WNC processing [6]. Because adaptive-array processing places sharp nulls in the directions of interferers, the presence of interferer motion does not provide sufficient nulling of the interferer given the number of snapshots available, which results in a masking of the desired target signal. Fig. 1 shows an example where source motion degrades the performance with 20 snapshots for a 128-element . array, especially on the weakest target at We also note that the bias of signal and noise has increased significantly due to source motion, which will be discussed in Section II. Null broadening allows the interferers to move through resolution cells while also being contained within a single wide null. In addition, the WNC can exploit the significant bias associated with snapshot deficiency [1]. The null-broadening concept [3]–[5] was originally developed to improve the robustness of the adaptive algorithms and demonstrated for a stationary problem. The potential of this approach, however, has not been fully explored due to its undesirable effects, such as decrease in array gain and broadening of the mainlobe. Here we extend the null-broadening approach to detect weak stationary targets in a nonstationary background such that only a limited number of snapshots are available due to fast-moving strong interferers crossing many resolution cells. Specifically, in this article we • review adaptive planewave beamforming vis a vis snapshot and bias issues; • describe the null-broadening techniques in terms of eigenvalues; • demonstrate the robustness of the null-broadening approach combined with the WNC processing for a snapshot-deficient problem arising from source motion in the presence of mismatch; • investigate the bias issues associated with the processing method; • characterize the performance of the null-broadening approach using probability of detection.
0364-9059/03$17.00 © 2003 IEEE
SONG et al.: NULL BROADENING WITH SNAPSHOT-DEFICIENT COVARIANCE MATRICES IN PASSIVE SONAR
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Fig. 2. Beampattern of linear N = 64 element array when steered broadside ( = 0) with interfering sources at u = 0:3 and u = 0:8: CBF (dashed line) and ABF (solid line). Note the two-deep nulls in the directions of the interferers with ABF.
(a)
where is the measured covariance matrix, is the steering vector pointing degrees from the broadside, denotes the Hermitian transpose operation, and is the identity matrix. The optional diagonal loading of strength is included to control the white-noise gain. Fig. 2 shows the beampattern of a linear array with 64 sensors when steered to the broadwith half-wavelength spacing ( . The array is subjected to two stationary interfering side sources of the same amplitude and located at and . Note the deep and sharp nulls produced in the directions of two interferers with ABF (solid line) compared to a conventional beampattern in the background (dashed line). The interfering sources are 30 dB louder than the channel noise. The exact, full-rank CSDM without diagonal loading is used for this example such that (b) Fig. 1. Adaptive beamforming with K = 20 snapshots for a N = 128 element array: (a) 9 fixed sources and (b) 2 moving and 7 fixed sources. The source levels and positions are denoted by 3. The horizontal dashed line indicates the noise level minus the array gain (10 log N ). The effect of source motion over 20 snapshots is observable in (b), especially on the weakest target at u = sin = 0:7. Note that the bias of signal and noise has increased significantly due to source motion, which will be exploited in Section V.
0
(2) dB and dB. with A robust version of the MVDR beamformer is the white-noise gain-constraint (WNC) beamformer [6], which adjusts the diagonal loading for each steering angle to satisfy a white-noise such that constraint
II. SNAPSHOT-DEFICIENT PROCESSING We begin by briefly reviewing adaptive planewave beamforming (ABF). We then address snapshot-deficient processing due to source motion and discuss the bias issue and nulling of strong interferers.
(3) where is the number of elements of the array and is given by (1). In practice, the white-noise gain (WNG) is introduced as WNG
A. ABF MVDR adaptive beamforming places nulls in the direction of loud interferers in the acoustic environment described by the cross-spectral density matrix (CSDM) or covariance matrix [2]. The MVDR weights with diagonal loading is
(1)
dB
(4)
dB corresponds to a linear processor and where WNG dB corresponds to a pure MVDR processor. WNG dB will be used later in the simulations, which WNG is chosen as a compromise in the presence of mismatch in the array-element positions between the robustness of the conventional linear processor and the interference-rejection capability of the pure MVDR processor.
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B. Sample Covariance Matrix The sample covariance matrix is (5) are the complex Fourier-amplitude vectors of the where the receiver outputs at the frequency of interest and the th snapshot and is the number of snapshots. As discussed by Baggeroer and Cox [1], there are time and bandwidth limits on the number of snapshots available with large-aperture sonar arrays operating in a dynamic environment. At broadside, the mainlobe of a resolution cell has a cross-range extent of (6) where is the range to a source, is the aperture of the array, and is the wavelength. A source moving with tangential speed transits this resolution cell and is within the cell for duration (7) where is the bearing rate of the source. The limit to the available bandwidth for frequency averaging is determined by signals close to endfire. The estimate of the phase in the cross spectra is smeared when one averages over too large of a bandwidth. The available bandwidth is constrained by [1], [7] (8) , the transit time across the array at endwhere and gives the approximate number fire. The product of available. In this paper, we are primarily conof snapshots cerned with the case when source motion limits the number of snapshots assuming narrow-band signals. C. Snapshots and Bias The usual criterion employed in adaptive processing for adewas specified to be by Reed quate estimation of et al. [9]. This typically is unattainable for most sonar operating environments with multiple moving surface ships representing discrete sources, especially for large arrays with narrow beams. Carlson [11] suggested diagonal loading the sample covariance matrix to reduce the required samples to as few as 1-2 . Other results [2], [14], [15] suggest that effective nulling at least equal to twice the number of can be achieved with (i.e., for ). strong interfering sources When using a limited number of snapshots and diagonal loading, significant biases (loss in the estimated output power) are introduced in adaptive processing [1], [7], as seen earlier in Fig. 1. The Capon and Goodman formula for bias and variance , [20] is valid only for the case of no loading and with which is typically not the case for sonars. An analytical formula under some conditions on the is given in [15] for where denotes the diagonal loading (i.e., smallest interference eigenvalue).
Fig. 3. Mailloux approach [3] distributes a cluster of q equal-strength incoherent sources arranged in a line centered around each source direction with a trough width of W between the outermost nulls.
Since there are no analytical results for bias in general when with diagonal loading , Baggeroer and Cox [1] showed, via Monte Carlo simulations, two important features: 1) the bias does not depend upon direction and 2) the bias for is significant. In particular, the bias increases with a decrease in the number of snapshots. In the presence of mismatch, however, the bias depends upon direction such that strong signals are subject to much larger signal suppression than are weak signals [16]. It will be shown that the significant bias due to snapshot deficiency turns out to be beneficial because it can be exploited by the WNC processor, which can reduce the bias selectively resulting in a significant increase in dynamic range. The effect of diagonal loading on the bias (MVDR) is described theoretically using eigenanalysis in the Appendix, which confirms that the bias is independent of steering angle as indicated in [1]. III. NULL BROADENING In this section, we review the null-broadening approaches [3]–[5] with a focus on the useful property for a snapshot-deficient problem. The method is most simply presented by considering a line array, although it can be applied to two-dimensional planar arrays. A. Distribution of Fictitious Sources Assuming that the narrow-band signals impinging on the array are uncorrelated with each other as well as with the spatially white noise, the terms in the covariance matrix for a one-dimensional array are [3] (9) The sum is performed over all interfering sources with averaged for measured from power and direction cosines are the element locations, is the broadside. The numbers is a Kronecker delta function. the noise covariance, and in each of the inIn order to produce a trough of width terference directions , Mailloux [3] distributed a cluster of equal-strength incoherent sources around each original source, as shown in Fig. 3. In this case, the additional sources can be summed in closed form as a geometric sum and can be written as
(10)
SONG et al.: NULL BROADENING WITH SNAPSHOT-DEFICIENT COVARIANCE MATRICES IN PASSIVE SONAR
N
Fig. 4. Beampattern of a = 64 element array steered broadside with augmented covariance matrix ~ : Mailloux (dashed line) with q = 7 and Zatman (solid line). Two interfering sources are incident at u = 0:3 and u = 0:8. Note the null broadening obtained at these two locations using W = 0:1.
R
where and . Since there is no angle dependence in the sinc function, we obtain a new covariance matrix term (11) In this formulation, we have introduced a source strength rather than in [3]. equally distributed with level element In Fig. 4, the adaptive beampattern of a linear array is shown with the original covariance matrix of (2) replaced by the augmented covariance matrix in (11) with and (dashed line). As opposed to the sharp nulls in Fig. 2, the beampattern clearly shows null broadening. B. Dispersion Synthesis Rather than physically distributing fictitious sources, Zatman [4] used dispersion to widen the null of a narrow-band signal. centered Assuming a rectangular spectrum of bandwidth at frequency , the augmentation of the fictitious sources is achieved by a synthetic averaging of the narrow-band covariover the bandwidth ance matrix
(12) and is the where time delay between the elements. For actual broad-band signals, null broadening was demonstrated in [17] with experimental data by making use of waveguide invariant theory [18] and averaging the estimated array-covariance matrix across frequency. For a half-wavelength uniform line array , the wide-band covariance matrix can be calculated as the Hadamard (element-wise) product [19] of and as (13)
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Fig. 5. The eigenvalues of the original covariance matrix (crosses) and the tapered matrix ~ (circles) for an N = 64 element array. The significant number of eigenvalues has increased from 2 to 14. On the other hand, the larger eigenvalues have decreased, resulting from the CMT operation. The first five eigenvalues of the CMT matrix are also superimposed (squares).
R
T
where and cordefined in the Mailloux responds to half of the null width approach. The solid line in Fig. 4 shows the resulting beampattern using the wide-band covariance matrix with . It is interesting to note that the bandwidth implicitly for a fixed value of varies with the direction cosine to keep a constant. Although both approaches achieve null broadening to the desired width , note from Fig. 4 that the solid line produces flatter troughs in the adaptive pattern than does the dashed line. Zatman’s approach produces continuous fictitious sources dis, whereas the Mailloux aptributed along the beamwidth proach places a finite number of discrete sources within the beamwidth. As increases, the two approaches become identical. C. Covariance Matrix Taper Guerci [5] combined the above null-broadening approach with diagonal loading through the concept of a “covariance matrix taper” (CMT) and theoretically investigated the effect of CMT on the adaptive beampattern. In this paper, diagonal loading is handled separately by the robust WNC processor. The Mailloux–Zatman (MZ) null-broadening approach is described in (13) as a modification of the original sample covariance matrix through the CMT matrix , which is a positive semidefinite matrix with its diagonal entries equal to 1. Note that both and are, in general, positive semidefinite Hermitian matrices. Null broadening or the Hadamard operation increases the number of eigenvalues [degrees of freedom (DOF)] such that (14) whose proof can be found in [19] (Theorem 5.1.7). Fig. 5 demonstrates that the two eigenvalues corresponding to each interferer have increased to 14 above the noise level, since each interferer is represented by fictitious nearby sources. For
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resolution for this example. Note that corresponds to the number of eigenvalues larger than the noise level (or the effective rank of ) in Fig. 5 (circles) for each source. A lower bound on the angular resolution is derived in [23], applying the Cramer–Rao formalism demonstrating that it is proportional to ) and a factor depending on the the classical Rayleigh limit ( output signal-to-noise ratio (SNR). It is also shown in Fig. 6 that the CMT operation reduces the beamformer output power due to discrete sources since it distributes the source power over the null width . However, the reduction of the signal power is negligible as compared to the significant bias resulting from a small number of snapshots when applied to a snapshot-deficient problem, as discussed in Section II-C. Note that the total power is preserved since the trace of is not affected by the CMT matrix , whose diagonal entries are equal to 1. Accordingly, the largest eigenvalues of the original covariance (crosses) have decreased in (circles) in Fig. 5.
(a)
IV. NULL BROADENING WITH SNAPSHOT DEFICIENT
(b)
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Fig. 6. MVDR output power using the Mailloux approach (solid) with ~ : (a) = 5 and (b) = 7. The result with the Zatman approach is superimposed in the dashed line. Note that 7 produces resolvable discrete sources rather than a broad null. The dotted line is the output power with .
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